Finitistic dimension and restricted flat dimension

Abstract

We investigate the relations between finitistic dimensions and restricted flat dimensions (introduced by Foxby [L.W. Christensen, H.-B. Foxby, A. Frankild, Restricted homological dimensions and Cohen–Macaulayness, J. Algebra 251 (1) (2002) 479–502]). In particular, we show the following result. (1) If T is a selforthogonal left module over a left noetherian ring R with the endomorphism ring A, then rfd ( T A ) ⩽ fdim ( A A ) ⩽ id ( T R ) + rfd ( T A ) . (2) If T R is classical partial tilting, then fdim ( A A ) ⩽ fdim ( R R ) + rfd ( T A ) . (3) If A = A 0 ⊆ A 1 ⊆ ⋯ ⊆ A m = R are Artin algebras with the same identity such that, for each 0 ⩽ i ⩽ m − 1 , rad A i is a right ideal in A i + 1 and rfd ( A i + 1 A i ) < ∞ (e.g., A i + 1 A i is of finite projective dimension, or finite Gorenstein projective dimension, or finite Tor-bound dimension), then fdim ( R R ) < ∞ implies fdim ( A A ) < ∞ . As applications, we disprove Foxby's conjecture [H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004) 167–193] on restricted flat dimensions by providing a counterexample and give a partial answer to a question posed by Mazorchuk [V. Mazorchuk, On finitistic dimension of stratified algebras, arXiv:math.RT/0603179, 6.4].